What it does
A soil slope fails by sliding along a curved surface. This page treats that surface as a circle, cuts the soil above it into vertical slices, and compares the moment that resists sliding with the moment that drives it. The ratio is the factor of safety (FS): below 1 the slope fails.
- Bishop’s simplified method allows for the side forces between slices (horizontal only), so FS appears on both sides of its equation. The page iterates until the answer stops changing.
- The ordinary method of slices (Fellenius) ignores the side forces. It is simpler and usually gives a lower, more cautious FS, especially with high pore pressures.
- A grid search tries hundreds of circles to find the critical circle with the lowest FS, then refines it.
How to use it
- Give the slope height H and angle β. The toe sits at height 0, the crest at H, with level ground in front and behind.
- Give the soil: effective cohesion c′, friction angle φ′ and unit weight γ. Choose two soils to add a lower layer below a horizontal boundary (a weaker clay under a fill, say).
- Give the water: none, a pore pressure ratio ru (u = ru × the weight of soil above the slip surface per unit area), or a water table that runs straight from a level at the toe to a level behind the crest.
- Choose the critical circle to search a grid of centres, or one circle I give to check a circle from a report or a textbook. “Fit the grid to the slope” sets a sensible grid for the current height and angle.
The result shows both factors of safety, the critical circle drawn on the slope with its slices, and a slice table you can download as CSV.
Worked example
All numbers are made up. A 10 m high slope at 30° in soil with c′ = 10 kPa, φ′ = 25° and γ = 19 kN/m³, with ru = 0.25. The grid search tries about 1,200 circles and shows Bishop FS = 1.18 (ordinary 1.11). The critical circle passes through the toe with its centre near (1.83, 20.79) m and a radius of 20.87 m, and Bishop’s equation settles in 16 iterations. Without water the same slope gives FS ≈ 1.50, which shows how much pore pressure matters.
Good to know
- Watch the smallest m(α). When a circle exits steeply at the toe, m(α) gets small and Bishop’s method gives unreliable answers. The search skips circles with m(α) below 0.2.
- If the best centre is on the edge of the grid, the page tells you: widen the grid and run again.
- The search tries circles touching levels down to the deepest circle bottom. Put it at a hard layer if there is one; otherwise deep circles may be the critical ones.
- For study and planning, not a substitute for professional judgement. Real slopes need site data, other failure shapes and an engineer’s review.
FAQ
Why does Bishop give a higher FS than the ordinary method? The ordinary method’s assumption about forces between slices leaves out part of the friction on the slip surface. Bishop’s method is closer to more rigorous methods for circular surfaces.
What FS should I aim for? Many guides use about 1.3 to 1.5 for permanent slopes and lower values for short-term works, but the right value depends on the consequences of failure and your local code.
How many slices do I need? About 30 to 50 is plenty. The answer changes very little beyond that.
Where the numbers come from
- Method of slices for circular slip surfaces: Bishop's simplified method (Bishop 1955), FS = Σ[c′b + (W − ub)tan φ′]/m(α) ÷ ΣW sin α with m(α) = cos α + sin α tan φ′/FS, and the ordinary (Fellenius) method: general soil mechanics, written by fibub from the equations.
- Pore pressure ratio ru = u/(γh): Bishop and Morgenstern's definition, general soil mechanics.
- Example values are made up by fibub for illustration.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.